On the multivariate probability integral transformation

A general formula is given for computing the distribution function K of the random variable H(X,Y) obtained by taking the bivariate probability integral transformation (BIPIT) of a random pair (X,Y) with distribution function H. Of particular interest is the behavior of the sequence (Kn) corresponding to the BIPIT of pairs (Xn,Yn) of componentwise maxima Xn=max(X1,...,Xn) and Yn=max(Y1, ..., Yn) of random samples (X1,Y1),...,(Xn,Yn) from distribution H. Illustrations are provided and the potential for statistical application is outlined. Multivariate extensions are briefly considered.